Growing balanced covering sets

TL;DR

This paper demonstrates a method to sequentially select vertices in a bipartite graph to maintain balanced coverage across multiple blocks, extending Steinitz's lemma and addressing a question by Scott and Seymour.

Contribution

It introduces a new ordering technique for vertices in bipartite graphs to achieve balanced coverage across blocks under specific degree constraints.

Findings

Existence of an ordering with balanced block coverage within a bound

Extension of Steinitz's lemma to bipartite graph covering

Partially answers an open question by Scott and Seymour

Abstract

Given a bipartite graph with bipartition $(A,B)$ where $B$ is equipartitioned into $k\ge2$ blocks, can the vertices in $A$ be picked one by one so that at every step, the picked vertices cover roughly the same number of vertices in each of these blocks? We show that, if each block has cardinality $m$, the vertices in $B$ have the same degree, and each vertex in $A$ has at most $cm$ neighbors in every block where $c>0$ is a small constant, then there is an ordering $v_1,\ldots,v_n$ of the vertices in $A$ such that for every $j\in\{1,\ldots,n\}$, the numbers of vertices with a neighbor in $\{v_1,\ldots,v_j\}$ in every two blocks differ by at most $\sqrt{2(k-1)c}\cdot m$. This is related to a well-known lemma of Steinitz, and partially answers an unpublished question of Scott and Seymour.